State the Poincaré-Bendixson theorem.

A model of a chemical process obeys the second-order system

$$\dot{x}=1-x(1+a)+x^{2} y, \quad \dot{y}=a x-x^{2} y$$

where $a>0$. Show that there is a unique fixed point at $(x, y)=(1, a)$ and that it is unstable if $a>2$. Show that trajectories enter the region bounded by the lines $x=1 / q$, $y=0, y=a q$ and $x+y=1+a q$, provided $q>(1+a)$. Deduce that there is a periodic orbit when $a>2$.